where we adopt the Von-Neumann relative entropy [1]} for any two density operators \(\nu _1,\nu _0\in \mathcal {S}\)  as
\(S(\nu _1\Vert \nu _0) := \text{Tr}(\nu _1(\ln \nu _1-\ln \nu _0))\) , and \(\Pi ^*_1\)  is the solution to detector's optimization problem (REF ).
